Finite Group Databasebeta

← #29578820

Subgroups — #29578820

Order 15616.

558 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519765C211yesmaximal, characteristic
2#21445936C211yesmaximal, characteristic, complemented
3C2×#16839494C212yesmaximal, complemented
4C2×#16839491C212yesmaximal
5C61×#1744C211yesmaximal, characteristic, complemented
6C2×#14519765C411yescharacteristic
7C2×C4×Dic122C2211yescharacteristic
8C2×C4×D244C2211yescharacteristic
9C2×C4⋊#92418C2212yes—
10#16839668C411yescharacteristic
11#16839664C411yescharacteristic
12#16839653—22no—
13#16839628C411yescharacteristic
14C4×#11758198—22no—
15#16839517—22no—
16#16839494—24no—
17#16839491—24no—
18C2×C4×C8×C61C2212yes—
19C43×C61C2211yescharacteristic
20C61×#336—22no—
21C22×Dic122C2×C411yescharacteristic
22C22×D244C2×C411yescharacteristic
23C22×C4×D61—21no—
24C22×#92418—22no—
25#14519765—22no—
26#14519765—22no—
27C4×Dic122—22no—
28C4×Dic122—22no—
29C4×D244—22no—
30C4×D244—22no—
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×C4⋊Dic61C2×C411yescharacteristic
33C2×#11758196—21no—
34C2×#11758198—22no—
35C2×#11758198—22no—
36C2×#11758198—21no—
37C2×C4×Dic61—21no—
38C4⋊#92418—22no—
39C4⋊#92418—22no—
40#14519684—22no—
41#14519684—22no—
42C22×C8×C61—22no—
43C2×C42×C61C2×C411yescharacteristic
44C2×C42×C61C2×C411yescharacteristic
45C2×C42×C61D412yes—
46C2×C42×C61D412yes—
47C2×C42×C61C2311yescharacteristic
48C4×C8×C61—22no—
49C4×C8×C61D414yes—
50C23×D61—21no—
51C22×C4×C61C22⋊C412yes—
52C22×C4×C61C22⋊C412yes—
53C22×C4×C61C22×C411yescharacteristic
54C22×C4×C61C2×D411yescharacteristic
55C22×C4×C61C2×D411yescharacteristic
56C22×Dic61—21no—
57C22×Dic61—21no—
58C22×Dic61—21no—
59C2×D244—22no—
60C2×D244—22no—
61C2×D244—21no—
62C2×D244M1612yes—
63C2×C4×D61—41no—
64C2×C4×D61—41no—
65C2×C4×D61—44no—
66C2×Dic122—22no—
67C2×Dic122—22no—
68C2×Dic122—21no—
69C2×Dic122M1612yes—
70C2×C8×C61—24no—
71C2×C8×C61—44no—
72C42×C61—21no—
73C42×C61—21no—
74C42×C61—22no—
75C42×C61—24no—
76C42×C61—42no—
77C42×C61C2×D412yes—
78C42×C61#4112yes—
79#11758198—28no—
80#11758198—44no—
81#11758198—42no—
82C4⋊Dic61—22no—
83C4⋊Dic61—22no—
84#11758196—42no—
85C4×Dic61—41no—
86C4×Dic61—41no—
87C2×#92418—42no—
88C2×#92418—44no—
89C23×C61C2×C22⋊C411yescharacteristic
90C22×D61—22no—
91C22×D61—41no—
92C22×D61—41no—
93C22×D61—42no—
94C22×D61—42no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61C4×D412yes—
103C2×C4×C61#12112yes—
104C2×C4×C61#12411yescharacteristic
105C2×C4×C61#12711yescharacteristic
106C2×C4×C61C2×M1611yescharacteristic
107C2×C4×C61C2×M1611yescharacteristic
108C2×C4×C61#13112yes—
109C2×C4×C61#13514yes—
110C2×Dic61—22no—
111C2×Dic61—41no—
112C2×Dic61—41no—
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61—42no—
116C2×Dic61—42no—
117D244—24no—
118D244—42no—
119C4×D61—82no—
120C4×D61—82no—
121Dic122—24no—
122Dic122—42no—
123C8×C61—44no—
124#92418—82no—
125#9645—611nomaximal, Sylow
126C22×C61#39011yescharacteristic
127C22×C61#40711yescharacteristic
128C22×C61#40811yescharacteristic
129C22×C61#41612yes—
130C22×C61#42112yes—
131C2×D61—42no—
132C2×D61—44no—
133C2×D61—81no—
134C2×D61—81no—
135C2×D61—82no—
136C4×C61—22no—
137C4×C61—42no—
138C4×C61—42no—
139C4×C61#43512yes—
140C4×C61#43512yes—
141C4×C61#48812yes—
142C4×C61#49512yes—
143Dic61—42no—
144Dic61—81no—
145Dic61—81no—
146C4×#124—611no—
147#1831—611no—
148#1744D6111yescharacteristic, complemented
149C2×Q8⋊C8—612no—
150C2×D4⋊C8—612no—
151C2×C61#148212yes—
152C2×C61#148812yes—
153C2×C61#195511yescharacteristic
154C2×C61#195811yescharacteristic
155C2×C61#196311yescharacteristic
156D61—82no—
157C2×#124—611no—
158C2×C4×Q8—611no—
159C2×C4×D4—611no—
160C2×C4⋊C8—612no—
161C2×C4×C8C2×D6112yes—
162#391—611no—
163#390—611no—
164#388—1222no—
165#382—611no—
166C4×C22⋊C4—1222no—
167C43C2×D6111yescharacteristic
168#336—22no—
169#336—1222no—
170Q8⋊C8—1224no—
171D4⋊C8—1224no—
172C61#964511yescharacteristic, complemented, Sylow
173C22×Q8—611no—
174C22×D4—611no—
175C23×C4—1221no—
176C22×C8—22no—
177C22×C8—1222no—
178#124—1222no—
179#124—1222no—
180C4×Q8—1222no—
181C4×Q8—1222no—
182C4×D4—1222no—
183C4×D4—1222no—
184C2×C4⋊C4—611no—
185C2×C4⋊C4—611no—
186C2×C4⋊C4—1221no—
187C2×C22⋊C4—1222no—
188C2×C22⋊C4—1222no—
189C2×C22⋊C4—1221no—
190C2×C42—1221no—
191C2×C42C4×D6111yescharacteristic
192C2×C42C4×D6111yescharacteristic
193C2×C42D24412yes—
194C2×C42C22⋊D6112yes—
195C2×C42C22×D6111yescharacteristic
196C4⋊C8—1222no—
197C4⋊C8—1222no—
198C4×C8—22no—
199C4×C8D24414yes—
200#95—1222no—
201#95—1222no—
202C24—1221no—
203C2×Q8—612no—
204C2×Q8—1221no—
205C2×Q8—1222no—
206C2×Q8—1222no—
207C2×D4—612no—
208C2×D4—1222no—
209C2×D4—1221no—
210C2×D4—1222no—
211C22×C4—1221no—
212C22×C4—1221no—
213C22×C4—1221no—
214C22×C4—2441no—
215C22×C4—2441no—
216C22×C4—2444no—
217C22×C4#1175819812yes—
218C22×C4#1175819812yes—
219C22×C4C2×C4×D6111yescharacteristic
220C22×C4C2×D24411yescharacteristic
221C22×C4C2×C22⋊D6111yescharacteristic
222C2×C8—24no—
223C2×C8—44no—
224C2×C8—2442no—
225C2×C8—2444no—
226C4⋊C4—1222no—
227C4⋊C4—1222no—
228C4⋊C4—2442no—
229C22⋊C4—1228no—
230C22⋊C4—2444no—
231C22⋊C4—2442no—
232C42—21no—
233C42—21no—
234C42—22no—
235C42—24no—
236C42—42no—
237C42—2441no—
238C42—2441no—
239C42C2×D24412yes—
240C42#1175822212yes—
241C23—1222no—
242C23—2441no—
243C23—2442no—
244C23—2442no—
245C23—2441no—
246C23C2×#1175819811yescharacteristic
247Q8—1224no—
248Q8—2442no—
249D4—1224no—
250D4—2442no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—1222no—
259C2×C4—4882no—
260C2×C4—4882no—
261C2×C4—2442no—
262C2×C4—2442no—
263C2×C4—2441no—
264C2×C4—2441no—
265C2×C4—2441no—
266C2×C4—2441no—
267C2×C4C4×D24412yes—
268C2×C4#1451975112yes—
269C2×C4#1451976511yescharacteristic
270C2×C4#1451977411yescharacteristic
271C2×C4M16×D6111yescharacteristic
272C2×C4M16×D6111yescharacteristic
273C2×C4#1451978512yes—
274C2×C4#1451979914yes—
275C8—44no—
276C8—4882no—
277C22—4882no—
278C22—4881no—
279C22—4881no—
280C22—2444no—
281C22—2442no—
282C22#1683966411yescharacteristic
283C22#1683972511yescharacteristic
284C22#1683973011yescharacteristic
285C22#1683975912yes—
286C22#1683978012yes—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—4881no—
291C4—4881no—
292C4—2442no—
293C4#1683983512yes—
294C4#1683983512yes—
295C4#1684011712yes—
296C4#1684014612yes—
297C2—4882no—
298C2#2144422812yes—
299C2#2144425412yes—
300C2#2144661711yescharacteristic
301C2#2144663011yescharacteristic
302C2#2144665111yescharacteristic
subgroupG/H
centerC22×C4C2×D244
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1744C2
socleC23×C61C2×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1744D61
hypocenterC61#9645
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C43 ◁ #1744

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21445936C2semidirect
3. C2×#16839494C2semidirect
5. C61×#1744C2semidirect
148. #1744D61semidirect
172. C61#9645semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4×#145197652
#214459362
C2×#168394942
C2×#168394912
C61×#17442
#964561